Polytope of Type {24,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6}*1728a
Also Known As : {24,6}3. if this polytope has another name.
Group : SmallGroup(1728,12317)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 144, 432, 36
Order of s0s1s2 : 3
Order of s0s1s2s1 : 24
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {24,6}*576a
   4-fold quotients : {12,6}*432d
   9-fold quotients : {8,6}*192a
   12-fold quotients : {12,6}*144d
   16-fold quotients : {6,6}*108
   36-fold quotients : {4,6}*48b
   72-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      24 facets:
         12 of {12}*24
         12 of {24}*48
      72 vertex figures:
         72 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      20 facets:
         12 of {8}*16
         8 of {24}*48
      48 vertex figures:
         48 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2> of order 4.
      12 facets:
         6 of {12}*24
         6 of {24}*48
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      18 facets:
         12 of {6}*12
         6 of {24}*48
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         4 of {12}*24
         2 of {24}*48
         6 of {8}*16
      24 vertex figures:
         24 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      16 facets:
         12 of {4}*8
         4 of {24}*48
      24 vertex figures:
         24 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 14)(  6, 13)(  7, 16)(  8, 15)( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 46)( 22, 45)( 23, 48)( 24, 47)( 25, 33)( 26, 34)( 27, 35)( 28, 36)( 29, 38)( 30, 37)( 31, 40)( 32, 39)( 49,105)( 50,106)( 51,107)( 52,108)( 53,110)( 54,109)( 55,112)( 56,111)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 61,102)( 62,101)( 63,104)( 64,103)( 65,137)( 66,138)( 67,139)( 68,140)( 69,142)( 70,141)( 71,144)( 72,143)( 73,129)( 74,130)( 75,131)( 76,132)( 77,134)( 78,133)( 79,136)( 80,135)( 81,121)( 82,122)( 83,123)( 84,124)( 85,126)( 86,125)( 87,128)( 88,127)( 89,113)( 90,114)( 91,115)( 92,116)( 93,118)( 94,117)( 95,120)( 96,119);;
s1 := (  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 53)(  6, 54)(  7, 56)(  8, 55)(  9, 63)( 10, 64)( 11, 62)( 12, 61)( 13, 60)( 14, 59)( 15, 57)( 16, 58)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 70)( 23, 72)( 24, 71)( 25, 79)( 26, 80)( 27, 78)( 28, 77)( 29, 76)( 30, 75)( 31, 73)( 32, 74)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 95)( 42, 96)( 43, 94)( 44, 93)( 45, 92)( 46, 91)( 47, 89)( 48, 90)( 99,100)(103,104)(105,111)(106,112)(107,110)(108,109)(115,116)(119,120)(121,127)(122,128)(123,126)(124,125)(131,132)(135,136)(137,143)(138,144)(139,142)(140,141);;
s2 := (  2,  4)(  5, 16)(  6, 13)(  7, 14)(  8, 15)( 10, 12)( 17, 33)( 18, 36)( 19, 35)( 20, 34)( 21, 48)( 22, 45)( 23, 46)( 24, 47)( 25, 41)( 26, 44)( 27, 43)( 28, 42)( 29, 38)( 30, 39)( 31, 40)( 32, 37)( 49, 65)( 50, 68)( 51, 67)( 52, 66)( 53, 80)( 54, 77)( 55, 78)( 56, 79)( 57, 73)( 58, 76)( 59, 75)( 60, 74)( 61, 70)( 62, 71)( 63, 72)( 64, 69)( 82, 84)( 85, 96)( 86, 93)( 87, 94)( 88, 95)( 90, 92)( 97,129)( 98,132)( 99,131)(100,130)(101,144)(102,141)(103,142)(104,143)(105,137)(106,140)(107,139)(108,138)(109,134)(110,135)(111,136)(112,133)(114,116)(117,128)(118,125)(119,126)(120,127)(122,124);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(144)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 14)(  6, 13)(  7, 16)(  8, 15)( 17, 41)( 18, 42)( 19, 43)( 20, 44)( 21, 46)( 22, 45)( 23, 48)( 24, 47)( 25, 33)( 26, 34)( 27, 35)( 28, 36)( 29, 38)( 30, 37)( 31, 40)( 32, 39)( 49,105)( 50,106)( 51,107)( 52,108)( 53,110)( 54,109)( 55,112)( 56,111)( 57, 97)( 58, 98)( 59, 99)( 60,100)( 61,102)( 62,101)( 63,104)( 64,103)( 65,137)( 66,138)( 67,139)( 68,140)( 69,142)( 70,141)( 71,144)( 72,143)( 73,129)( 74,130)( 75,131)( 76,132)( 77,134)( 78,133)( 79,136)( 80,135)( 81,121)( 82,122)( 83,123)( 84,124)( 85,126)( 86,125)( 87,128)( 88,127)( 89,113)( 90,114)( 91,115)( 92,116)( 93,118)( 94,117)( 95,120)( 96,119);
s1 := Sym(144)!(  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 53)(  6, 54)(  7, 56)(  8, 55)(  9, 63)( 10, 64)( 11, 62)( 12, 61)( 13, 60)( 14, 59)( 15, 57)( 16, 58)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 70)( 23, 72)( 24, 71)( 25, 79)( 26, 80)( 27, 78)( 28, 77)( 29, 76)( 30, 75)( 31, 73)( 32, 74)( 33, 81)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 95)( 42, 96)( 43, 94)( 44, 93)( 45, 92)( 46, 91)( 47, 89)( 48, 90)( 99,100)(103,104)(105,111)(106,112)(107,110)(108,109)(115,116)(119,120)(121,127)(122,128)(123,126)(124,125)(131,132)(135,136)(137,143)(138,144)(139,142)(140,141);
s2 := Sym(144)!(  2,  4)(  5, 16)(  6, 13)(  7, 14)(  8, 15)( 10, 12)( 17, 33)( 18, 36)( 19, 35)( 20, 34)( 21, 48)( 22, 45)( 23, 46)( 24, 47)( 25, 41)( 26, 44)( 27, 43)( 28, 42)( 29, 38)( 30, 39)( 31, 40)( 32, 37)( 49, 65)( 50, 68)( 51, 67)( 52, 66)( 53, 80)( 54, 77)( 55, 78)( 56, 79)( 57, 73)( 58, 76)( 59, 75)( 60, 74)( 61, 70)( 62, 71)( 63, 72)( 64, 69)( 82, 84)( 85, 96)( 86, 93)( 87, 94)( 88, 95)( 90, 92)( 97,129)( 98,132)( 99,131)(100,130)(101,144)(102,141)(103,142)(104,143)(105,137)(106,140)(107,139)(108,138)(109,134)(110,135)(111,136)(112,133)(114,116)(117,128)(118,125)(119,126)(120,127)(122,124);
poly := sub<Sym(144)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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